A novel finite element method, as an alternative to upwinding, is prop
osed based on the elimination of the factors which could cause numeric
al oscillation and instability by properly choosing a set of unconvent
ional weighting functions. The proposed method is first developed and
verified for a one dimensional case and then extended to two dimension
al problems. The calculation results for a 2D problem, along with the
exact solutions and those obtained from Galerkin's and ''optimal' upwi
nding methods, show that the proposed method is superior to the other
two methods in terms of accuracy and freedom from oscillation.